Enumeration in stochastic processes and polyhedral geometry

Abstract

This dissertation explores the combinatorics of Markov chains and polyhedral geometry, with a focus on the asymmetric simple exclusion process (ASEP) and the Ehrhart theory of polytopes. The first part addresses the stationary distribution of stochastic models, including the open ASEP, the Arndt-Heinzel-Rittenberg (AHR) model and the doubly ASEP (DASEP). We give a two-layer simple random walk interpretation for the open ASEP model, a tableaux formula for the AHR model, and show that the DASEP exhibits homomesy phenomenon. The second part of the dissertation studies the Ehrhart theory of positroid polytopes and alcoved polytopes. We present combinatorial formulas for the $h^*$-polynomials of positroid polytopes and alcoved polytopes. We also prove a connection between our shelling formula with decorated ordered set partitions.